Featured Researches

Classical Analysis And Odes

A d -dimensional Analyst's Travelling Salesman Theorem for general sets in R n

In his 1990 paper, Jones proved the following: given E⊆ R 2 , there exists a curve Γ such that E⊆Γ and H 1 (Γ)∼diamE+ ∑ Q β E (3Q ) 2 ℓ(Q). Here, β E (Q) measures how far E deviates from a straight line inside Q . This was extended by Okikiolu to subsets of R n and by Schul to subsets of a Hilbert space. In 2018, Azzam and Schul introduced a variant of the Jones β -number. With this, they, and separately Villa, proved similar results for lower regular subsets of R n . In particular, Villa proved that, given E⊆ R n which is lower content regular, there exists a `nice' d -dimensional surface F such that E⊆F and H d (F)∼diam(E ) d + ∑ Q β E (3Q ) 2 ℓ(Q ) d . In this context, a set F is `nice' if it satisfies a certain topological non degeneracy condition, first introduced in a 2004 paper of David. In this paper we drop the lower regularity condition and prove an analogous result for general d -dimensional subsets of R n . To do this, we introduce a new d -dimensional variant of the Jones β -number that is defined for any set in R n .

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Classical Analysis And Odes

A β -Sturm Liouville problem associated with the general quantum operator

Let I?�R be an interval and β:I?�I a strictly increasing and continuous function with a unique fixed point s 0 ?�I which satisfies ( s 0 ?�t)(β(t)?�t)?? for all t?�I , where the equality holds only when t= s 0 . The general quantum operator defined in 2015 by Hamza et al., D β [f](t):= f(β(t))?�f(t) β(t)?�t if t??s 0 and D β [f]( s 0 ):= f ??( s 0 ) if t= s 0 , generalizes the Jackson q -operator D q and also the Hahn (q,?) -operator, D q,? . Regarding a β??Sturm Liouville eigenvalue problem associated with the above operator D β , we construct the β??Lagrange's identity, show that it is self-adjoint in L 2 β ([a,b]), and exhibit some properties for the corresponding eigenvalues and eigenfunctions.

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Classical Analysis And Odes

A Douglas-Rachford construction of non-separable continuous compactly supported multidimensional wavelets

After re-casting the n -dimensional wavelet construction problem as a feasibility problem with constraints arising from the requirements of compact support, smoothness and orthogonality, the Douglas--Rachford algorithm is employed in the search for one- and two-dimensional wavelets. New one-dimensional wavelets are produced as well as genuinely non-separable two-dimensional wavelets in the case where the dilation on the plane is the standard D a f(t)= a −1 f(t/a) (t∈ R n ,a>0) .

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Classical Analysis And Odes

A Generalized Mountain Pass Lemma with a Closed Subset for Locally Lipschitz Functionals

The classical Mountain Pass Lemma of Ambrosetti-Rabinowitz has been studied, extended and modified in several directions. Notable examples would certainly include the generalization to locally Lipschitz functionals by K.C. Chang, analyzing the structure of the critical set in the mountain pass theorem in the works of Hofer, Pucci-Serrin and Tian, and the extension by Ghoussoub-Preiss to closed subsets in a Banach space with recent variations. In this paper, we utilize the generalized gradient of Clarke and Ekeland's variatonal principle to generalize the Ghoussoub-Preiss's Theorem in the setting of locally Lipschitz functionals. We give an application to periodic solutions of Hamiltonian systems.

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Classical Analysis And Odes

A Gneiting-Like Method for Constructing Positive Definite Functions on Metric Spaces

This paper is concerned with the construction of positive definite functions on a cartesian product of quasi-metric spaces using generalized Stieltjes and complete Bernstein functions. The results we prove are aligned with a well-established method of T. Gneiting to construct space-time positive definite functions and its many extensions. Necessary and sufficient conditions for the strict positive definiteness of the models are provided when the spaces are metric.

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Classical Analysis And Odes

A New Inequality For The Hilbert Transform

Suppose that { a j }∈ l 1 and a j ≠0 for all j . Then we prove that there is a constant C such that ∑ n=1 ∞ ♯{k∈Z: ∣ ∣ ∣ ∣ ∑ i=−n n ′ a k+i i ∣ ∣ ∣ ∣ >λ}≤ C λ ∑ i=−∞ ∞ | a i | for all λ>0 . We show as a corollary that one can use a transference argument to have an analogue result for the ergodic Hilbert transform.

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Classical Analysis And Odes

A New Linear Inversion Formula for a class of Hypergeometric polynomials

Given complex parameters x , ν , α , β and γ∉−N , consider the infinite lower triangular matrix A(x,ν;α,β,γ) with elements A n,k (x,ν;α,β,γ)=(−1 ) k ( n+α k+α )⋅F(k−n,−(β+n)ν;−(γ+n);x) for 1⩽k⩽n , depending on the Hypergeometric polynomials F(−n,⋅;⋅;x) , n∈ N ∗ . After stating a general criterion for the inversion of infinite matrices in terms of associated generating functions, we prove that the inverse matrix B(x,ν;α,β,γ)=A(x,ν;α,β,γ ) −1 is given by B n,k (x,ν;α,β,γ)= (−1 ) k ( n+α k+α )⋅ [ γ+k β+k F(k−n,(β+k)ν;γ+k;x)+ β−γ β+k F(k−n,(β+k)ν;1+γ+k;x)] for 1⩽k⩽n , thus providing a new class of linear inversion formulas. Functional relations for the generating functions of related sequences S and T , that is, T=A(x,ν;α,β,γ)S⟺S=B(x,ν;α,β,γ)T , are also provided.

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Classical Analysis And Odes

A Nonlinear Variant of Ball's Inequality

We adapt a recent induction-on-scales argument of Bennett, Bez, Buschenhenke, Cowling, and Flock to establish a global near-monotonicity statement for the nonlinear Brascamp-Lieb functional under a certain heat-flow, from which we establish some finiteness and stability results for the associated nonlinear Brascamp-Lieb inequality.

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Classical Analysis And Odes

A Note on Non-tangential Convergence for Schrödinger Operators

The goal of this note is to establish non-tangential convergence results for Schrödinger operators along restricted curves. We consider the relationship between the dimension of this kind of approach region and the regularity for the initial data which implies convergence. As a consequence, we obtain a upper bound for p such that the Schrödinger maximal function is bounded from H s ( R n ) to L p ( R n ) for any s> n 2(n+1) .

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Classical Analysis And Odes

A Note on the Axisymmetric Diffusion equation

We consider the explicit solution to the axisymmetric diffusion equation. We recast the solution in the form of a Mellin inversion formula, and outline a method to compute a formula for u(r,t) as a series using the Cauchy residue theorem. As a consequence, we are able to represent the solution to the axisymmetric diffusion equation as rapidly converging series.

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